I really enjoyed this one. It took me two nights because I got stuck with all of the 5s and 6s on the left. I had figured out that they connect along the entire left edge of the figure, but it took me a while to break where each 5 went after that. I really had to focus on how to get the top two blank nodes to connect in a way that all of the puzzle would be connected.

The two blank nodes are meant to be connected, as you can see from the dotted line. I'm sure Cauchy only represented it that way out of uncertainty. The disconnected node can only connect to its neighbor.

Ah, I didn't see the dashed line at all - currently awaiting eye surgery. I was counting the bottom edge as one segment, which is why my "solution" didn't match at all - if I'd drawn it with two lines rather than one I'd have counted differently.

CharlieP wrote:Ah, I didn't see the dashed line at all - currently awaiting eye surgery. I was counting the bottom edge as one segment, which is why my "solution" didn't match at all - if I'd drawn it with two lines rather than one I'd have counted differently.

If you were counting the bottom edge as one segment, then you shouldn't have been able to find a solution at all.

Spoiler:

First, the two nodes at the bottom have to be connected to each other. Then they both must connect to the 8 and the 6. This is due to rule 2. Without a vertex in the center, this makes a path of length three leading to the 6, so another path of length three needs to connect to the 6 as well (or a combination of paths that sum to length three), because of rule 3.

It can't connect to either of the two nodes on the right, because that length is only two. The only two nodes of distance three are the 6 in the center of the puzzle and the O next to the 8 in the bottom left. To get to either of those nodes, the segment path from the 6 has to pass through the center vertex of the bottom hexagon.

Now the 8 on the left still needs segments that sum to length five to connect to it. The only paths available are to the O node next to it (distance one) and the vertex at the center of the hexagon. You can see that no matter what, it must connect to the vertex in the center - but that vertex must also connect to the 6 on the right, which is already connected to the 8 along the bottom. This creates a loop, which violates rule 4.

If you were able to get any solution past that, then I wonder if you misunderstood one or more of the rules.

CharlieP wrote:Ah, I didn't see the dashed line at all - currently awaiting eye surgery. I was counting the bottom edge as one segment, which is why my "solution" didn't match at all - if I'd drawn it with two lines rather than one I'd have counted differently.

If you were counting the bottom edge as one segment, then you shouldn't have been able to find a solution at all.

Spoiler:

First, the two nodes at the bottom have to be connected to each other. Then they both must connect to the 8 and the 6. This is due to rule 2. Without a vertex in the center, this makes a path of length three leading to the 6, so another path of length three needs to connect to the 6 as well (or a combination of paths that sum to length three), because of rule 3.

It can't connect to either of the two nodes on the right, because that length is only two. The only two nodes of distance three are the 6 in the center of the puzzle and the O next to the 8 in the bottom left. To get to either of those nodes, the segment path from the 6 has to pass through the center vertex of the bottom hexagon.

Now the 8 on the left still needs segments that sum to length five to connect to it. The only paths available are to the O node next to it (distance one) and the vertex at the center of the hexagon. You can see that no matter what, it must connect to the vertex in the center - but that vertex must also connect to the 6 on the right, which is already connected to the 8 along the bottom. This creates a loop, which violates rule 4.

If you were able to get any solution past that, then I wonder if you misunderstood one or more of the rules.

No, the scare quotes around "solution" were deliberate - I was able to continue with a mistake in place for a surprisingly long time, until I realised, with most of the lines drawn in, that it wasn't going to work. I then tried working backwards to find where I'd gone wrong, getting more and more confused, before giving up and looking at the solution to see if I was even close. I should have probably just started again.